How to apply Noise texture to specific portion?
Connect your Color Ramp output also to Math: Subtract with 1 set as the first value - this way you invert your value, where black is represented by 1, white by 0 and you keep the gradient (smooth transition). Now multiply Parabola Math custom node by that inverted value to, so that black areas are multiplied by 1 (the height doesn't change), white areas are multiplied by 0 (the height becomes 0) and areas in between are multiplied by a fraction (get higher and higher towards the black).

In linear interpolation, the height change is too sudden, so you might want to switch it to Ease:

For a finer control, you might want to use another color ramp to control the displacement transition specifically (this way you don't have to change the color by using Ease mode on color going to the shader), though I wouldn't recommend moving the sliders on the 2nd Color Ramp, I was getting ugly results...